Chris Wilcox - Thesis - Ch 2
Objects and Indispensability
As mentioned briefly earlier, When attempting to criticise the indispensability argument, one obvious (and popular) strategy is to question the truth of one or more of its premises (indeed, Chapter 3 will focus largely on two criticisms of this type). However, I believe that there is another viable, and heretofor overlooked, critical strategy. Namely, one may argue that it seems (at the very least) highly questionable that the indispensability argument can produce the kind of “mathematical” entities that will satisfy either the Platonist or the IBE-based scientific realist (hereafter referred to simply as the “scientific realist”). Accordingly, in this chapter I will not question the truth of the premises of the indispensability argument (or indeed their supporting background assumptions, those that are explicitly stated at least). I will simply assume the truth of these results with a view to investigating exactly what one can get out of the indispensability argument. I believe that when one compares the result of this investigation to what would commonly be expected of mathematical objects by Platonists and scientific realists, the entities produced by the indispensability argument are found wanting. More problematically, they are found wanting in a way that is likely detrimental to mathematical and scientific pratice and thus is at tension with the naturalist urge. Following on from this conclusion, I will argue that at the very least more work needs to be done before the proponents of indispensability can claim to have produced “mathematical objects” in a sense that is acceptable to most stakeholders in the debate. I will begin by explicitly outlining just which parts of Colyvan’s argument I will be assuming to be true.
Indispensability
Firstly (and most obviously), I will be assuming that all of the background Quinean doctrines are true. That is, I will assume that theories are confirmed or disconfirmed as wholes (confirmational holism); that we should look to science (in particular our best scientific theories) in order to determine the nature of the world (the normative strand of Quinean naturalism); and that the ontological commitments of our best scientific theories are to be ascertained by determining the domain of quantification of the bound variables involved in the canonical logical representations of those theories (the Quinean ontic thesis). Furthermore, I will take the combined truth of these assertions to be adequate to prove that we should believe in those entities that are indispensable to our best scientific theories. Additionally, I will assume for the sake of argument (contra Field, see \citet{Field:1980 and :1989 for details here.}) that mathematical entities are indeed indispensable for the formulation of our best scientific theories. Finally, I will also assume that the scientific realist is, by virtue of their scientific realist commitments, also committed to the indispensability argument for Platonism in mathematics (i.e. that acceptence of IBE-based realism implies acceptance that the indispensability argument produces “real” objects). As stated above, my claim in this chapter is that even assuming all this, it is highly questionable whether the indispensability argument should convince anyone of the existence of mathematical objects. To begin to see this, one must first examine certain commonly held views regarding the nature of mathematical entities.
is a Mathematical Entity?
As indicated above, in order to see why the indispensability argument is likely not attractive to either the scientific realist or the Platonist (at least not in its present state) one must examine the nature of mathematical entities in more detail. By nature here, I primarily mean the attributed to such entities. Indeed, there is just one particular property (or group of properties) upon which I wish to focus my investigation: those conveying abstractness (or something akin to it) upon mathematical entities. Hereafter I shall refer to these properties as “abstractness properties.”
Now, as Colyvan notes, what it means exactly for a mathematical object to be abstract is quite a difficult and unsettled issue amongst Platonists. Indeed, what it is more generally for an object or entity to be abstract is quite a difficult and unsettled issue in analytic philosophy. However, as discussed briefly in the previous chapter, I believe that Colyvan’s response to this difficulty is unsatisfactory. In essence it seems that he deals with the issue by simply setting aside discussion of the abstractness properties of mathematical objects, choosing to focus instead on the properties that are highlighted by their involvement in scientific theories. While the issue of a mathematical entity’s mathematical properties is indeed an important one, in setting aside the problematic abstractness properties Colyvan has denied the scientific realist precisely the sorts of things they would likely point to in order to justify their anti-Platonist stance. Furthermore, he has also provided an account of the nature of mathematical objects that most Platonists would probably hold to be too limited.
But just what does it mean for a mathematical object to be abstract? It will be useful to get feel for this issue by exploring a range of options via which the abstractness desiderata may be satisfied.
As Rosen observes~\footnote there is as yet no standard philosophical account of the difference between abstract and concrete objects. However, as he also points out, if anything is to be considered an abstract object, then it is generally acknowledged that mathematical objects should be so considered. Thus, if some sort of abstractness property exists (and it is quite a widely held view that it does), then mathematical objects are highly probable candidates for possessing it.
There are numerous potential methods via which the abstract/concrete distinction may be made. However, given that my primary concern here is with the abstractness of mathematical objects more generally conceived, I will limit my discussion here to a brief sketch of a few of the more popular options (so as to convey a general idea of the sorts of properties commonly subsumed under the heading of abstractness). One popular claim is that abstract objects are abstract in the sense of being non-spatiotemporal (i.e. the claim is that they exist outside of space and time). Alternatively, abstract objects may be held to be abstract in the sense of being causally inert (i.e. that they abstract because they unable to causally interact with the physical world). Finally, abstract objects may be determined to be abstract in the sense of being the result (or insight) of a process of abstraction applied to concrete objects. Such processes have been proposed by Locke and, more recently, Hale and Wright in the form of a neo-Fregean conception. Its hould be noted that such a process should not necessarily be interpreted as the process via which such objects are brought into existence, rather it may just be that the abstraction is the process via which we come to intuit the existence of abstract objects. Additionally, it should be acknowledged that all of the conceptions discussed have been subjected to serious criticism. Given this, any conception of abstract mathematical objects involving them should be considered to be controversial.
However, there is a possible way of addressing the common intuition behind the abstractness desideratum in a more minimal way: namely, via the claim that mathematical objects should be multiply instantiable — i.e. they should be types rather than tokens. Indeed, it may well be that it is this aspect of the nature of mathematical entities that gives rise to the urge to posit them as abstract in the first place.
By multiple instantiability here I mean that the mathematical entity (consider a particular Abelian group for example) should be able to be posited to exist instantiated in one theory here and in another theory there. A useful analogy can be drawn here to a more tangible scientific object type such as a hydrogen atom. According to most theories, the hydrogen atom type is held to be multiply-instantiated in various molecules (i.e. the molecules involve different tokens of the same type). However, this analogy of hydrogen atoms and mathematical objects fails in an important and illuminatng way: with regard to the level at which their existences is posited. In the case of the hydrogen atom, it is generally held that only the particular hydrogen atom tokens exist, not the hydrogen atom type; whereas in the mathematical case, type-level existence is exactly what many (if not most) Platonists wish to claim was the case. Additionally, it seems open to the scientific realist to claim that the type level is the only level at which it is feasible to posit the existence of mathematical entities (more on this later). Again, these claims could not be said to be uncontroversial (although they are almost certainly less controversial than any of the other particular abstractness claims above). However, it does help one to develop a feel for the range of ways in which the abstractness desiderata may be satisfied. Also, I will argue below that any account of mathematical objects that does not at least account for their multiple instantiability runs into serious problems with regard to naturalist considerations.
What has begun to emerge from the discussion above is a sort of cline~cline is an axis that is not necessarily a continuum - a more accurate model for many philosophical debates. along which one may place the various ways in which mathematical objects may satisfy the abstractness desiderata. The cline imposes an order upon these methods of satisfaction in terms of the level of controversy one would court by holding such a view: from fully-fledged abstractness properties such as causal inertness at the more controversial end of the axis to multiple-instantiability towards the other. What I argue that Colyvan must do (and has not done) in order to out forward a position that at least has a chance of satisfying the Platonist and causing problems for the scientific realist, is place the mathematical objects resultant from the indispensability argument somewhere on his cline. Indeed, it is not at all clear that the Quinean Ontological Machine (again abreviated from here onwards as the “QOM”) provides him with the tools necessary to do so.
Can the Quinean Ontological Machine Deliver?
As noted in the previous chapter, when an entity is reified it is by no means certain which (if any) of its properties are also thereby determined to be real. To put it another way: not all of an object’s properties need be held to be properties of that object (recall the colour example of Chapter 1). Accordingly then, even if one has successfully argued for the existence of a particular type of entity, one needs to give some sort of principled account of which of that object’s properties are to be held to be real too (in virtue of their being intrinsic for example). In what follows I will argue that it is hard to see how Colyvan can give such an account that is a) in accordance with the principles underlying the indispensability argument; and b) able to explain in a principled way how mathematical objects have certain (real) properties many (if not most) would wish them to have (in particular, properties).
Indispensability
The most obvious way in which Colyvan might attempt to provide such an account is by extending his basic criterion for ontological commitment at the entity level, i.e. indispensability to our best scientific theories (further possible strategies will be considered below). How might such an account work? According to this strategy then, an object’s real properties would be just those that are indispensable to its involvement in our best scientific theories. However, this doesn’t seem to deliver the required abstractness properties. It is hard to see how such properties play a role when one considers mathematical entities qua their role in our best scientific theories (although they may be at work in the of such theories — this possibility will be discussed later). Indeed, this strategy likely doesn’t even get one many of the usual, properties of objects. Ordinarily, a mathematical object consider qua mathematical object is considered to have a multitude of mathematical properties (with more being discovered in mathematics departments all the time). However, clearly not all of these are the result of involvement in our best scientific theories. Indeed, it would be extremely unusual for this set of mathematical properties to be identical with the set of those properties resultant from considering a mathematical object qua its involvement in our best scientific theories. Furthermore, it is certainly not the case that these two sets would be identical in all cases. For example, consider the group formed by the vectors in a Minkowski space together with the usual vector addition (validated here by its involvement in the equations of the Lorentz transformations). Mathematically, such a group would be considered to have many properties (such as its number of subgroups) that were not indispensable to the role of Minkowski space in special relativity. Again, the consequences of this incongruence between what is delivered by the QOM and mathematical practice will be explored further below.
Essentially, the most important thing that the investigation of the preceding paragraph has highlighted is that, according to this “extension of indispensability” interpretation of which properties should be held to exist, one of the crucial properties that the QOM doesn’t deliver is multiple instantiability (or indeed any kind of abstractness). If it did deliver such a property, then Colyvan could possibly distinguish between those mathematical properties essential to the of object and that subset involved in its scientific application with a view to claiming that the (potentially larger) set of those essential to the type should be considered real. Instead, the mathematical objects resultant from Colyvan’s argument under this interpretation do not exhibit a type-token structure at all, rather their “real” properties are completely determined by, and inseparable from, the context of the instances in which they are reified: the context of particular scientific theories. These objects then, are far more akin to the hydrogen atoms of my earlier analogy in that their existence is not posited at the type level. Mathematical objects so conceived are tokens without a type, they are particulars not universals. Furthermore, the difficulties of deriving abstractness properties using the QOM discussed above suggests that they cannot even be conceived of as particulars (i.e. tropes). Rather, they should be seen as concrete particulars.
Indispensability and Scientific Practice
Okay, one might say, under Colyvan’s conception mathematical objects are concrete particulars that do not actually have all of the properties (mathematical or otherwise) that many have believed them to. What’s the problem with that? Perhaps what results from the indispensability argument is a more accurate picture of just what there is in the world, one that may help us to overcome the prejudices of past practice. Accordingly to this response, we just need to bite the bullet and accept this new picture of the world. However, there seem to be at least two possible (related) responses to such a position: firstly, one may (following one interpretation of Wittgenstein) make the general claim that philosophical investigations should not be revisionist when it comes to current social practice; secondly (and more viably), one may claim that some revisionism is okay, but not when the resultant position would be detrimental to scientific progress.~approach owes a great deal to Price’s doctrine of subject naturalism, see The first claim is highly controversial and probably too general to be of much use. For example, how does one overcome ny sort of entrenched dogma given the restrictions it places upon philosophical endeavour? The second claim seems far more plausible and indeed appears to be something that should supported by many, if not all naturalists. Accordingly then, I will focus my analysis on how one might apply the second claim here, arguing that the revisionism of Colyvan’s position (as currently conceived anyway) is indeed detrimental to scientific progress. The problem is that mathematicians (including applied mathematicians) do not (and in many cases ) work with concrete particulars. Many, if not most, mathematical investigations are conducted at the level of mathematical object . For example, consider the incoherence of the notion of an isomorphism posited at the level of tokens rather than types.~am endebted to Jason Grossman for this example. Given this, and the centrality of mathematical investigations to scientific endeavour, it seems at the very least highly plausible to claim that Colyvan’s position would be detrimental to science and that as such, the indispensability argument shouldn’t been accepted.
Versions of the Quinean Ontological Machine
The considerations of the previous paragraph seem to have highlighted that any account of mathematical objects must at least allow them to be conceived of as types rather than tokens (if one wants to satisfy the naturalist urge at least). Furthermore, it seems that on the most obvious extension of the QOM to give an account of which properties should be considered real, this is not possible. But perhaps there are other extension strategies that Colyvan may employ in order to meet this challenge? I will now consider two such potential strategies: one involving the claim that mathematical practice should itself be priviledged in some sense; and another invoking the claim that indispensability should only really be posited at the level of objects and that some other story needs to be told regarding their properties. As we shall see, both of these responses have their problems.
Under the first strategy, an argument could be mounted the effect that mathematical also obtains some sort of privileged status as a result of the involvement of mathematics in our best scientific theories. Hence, given that mathematical practice is at least committed to abstractness properties in the sense of multiple instantiability, the resultant mathematical objects do indeed satisfy the abstractness desiderata (although it is still hard to see how one could get an abtractness property stronger than multiple instantiability using this strategy).
But does this really gel with the Quinean principles underpinning the indispensability argument? It seems, at least superficially, that when we privilege mathematics, we move away from the idea of science as gaining a priviliged status resultant from its being the way we investigate the world. That is, unless we just hold mathematics to a form of scientific endeavour. But if this is the case, then the indispensabilty argument as presently conceived seems a very roundabout way of arguing for Platonism. If one holds this view, there is a much more direct form the argument could take: {[green NEED A LIST HERE ]} - Premise 1. We should believe in things that are indispensable to our best scientific practice. - Premise 2. Mathematics is part of our best scientific practice. - Premise 3. Mathematical entities are indispensable for mathematics. {[green THIS NEEDS TO BE A CONCLUSION LINE THINGAME ]}
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{[green LIST AGAIN HERE ]} - Conclusion. Therefore we should believe in mathematical entities.
But Premises 1 and . are certainly not uncontroversial assertions. Any proponent of the argument above would need to argue for them. Furthermore, even if one wishes to stick with the original indispensability argument, these two assertions appear to be the very same ones which must underwrite the claim that mathematical practice should be accorded a priviledged status made above. Additionally, these premises certainly do not straightforwardly follow from the Quinean underpinnings of the indispensability argument discussed in Chapter 1. Hence, the onus is on anyone wishing to argue for Platonism to justify these two principles. This has not been done.
The other alternative extension strategy involves somehow limiting the level at which indispensability considerations are relevant to that of objects and positing some other account of how one determines which properties are real. That is ontological commitment to objects would indeed be determined via indispensability, but ontological commitment to the of those objects would be determined via some other set of principles. There are at least two major issues any such account of the reification of properties needs to deal with. Firstly, many mathematical objects seem to be overdescribed. This can occur in two ways: firstly, a mathematical object may be differently described in different areas of mathematics — for example, using the terminology of point set topology or algebraic topology. While these differing descriptions are not imcompatible, are both to be validated by a mathematical object’s involvement in our best scientific theories? At best this seems to highlight an ambiguity in the argumen. Secondly, the description of a mathematical object (or even its existence) may vary depending on the particular set theory in use. For example, {[green E.G. FROM MADDY HERE ]}.~gives a very clear exposition of this in This kind of overdescription seems to be more problematic because in choosing to accept one set theoretic description one is generally ruling out the others and, as seen, this may involve a change in one’s mathematical ontology. At the very least, more detail is required of Colyvan here.
However, there is a more fundamental problem with the strategy of the preceding paragraph: it is just not clear how any such account can provide abstractness even in the sense of multiple instantiability. If indispensabilty is accepted as the means via which one comes to be ontologically committed to mathematical objects, then just what does that existence consist in if not in the reification of some set of properties. Any other account seems to be of such an irreducible nature as to be almost useless.~ Indeed, it would seem that the indispensability-based ontological criterion is at tension with any other further criterion that may be posited for properties. Furthermore, it is hard to see why the objects resultant from the indispensability criterion have not already been imbued with certain properties antithetical to multiple instantiability — namely, concreteness and particularity. In any event, it is obvious that at the very least Colyvan again owes us more of a story here.
I should make it clear at this point that I do not wish to claim that Colyvan is incapable of reponding to the challenges discussed above, only that a successful response has not been mounted yet and furthermore, that there seems to be no way of doing so. Hence, the onus is on Colyvan to respond either with an account of how abstractness properties (in particular multiple instantiability) are reified or with some justification as to why one should embrace the controversial nature of his position regarding the nature of mathematical entities. At present, Colyvan’s position seems quite a bit more controversial than it may have appeared on first inspection. If I am correct, then it is at tension with many versions of both Platonism and naturalism — both of which are camps in which he would hope to find many allies.
Despite the tentative nature of the conclusion above, even if Colyvan successfully mounts a response, the indispensability argument is not out of the woods. Several significant criticisms of the underlying assumptions of the argument can be made. In the next chapter I will focus on just these sorts of criticisms. Namely, those questioning the viability of Quinean naturalism.
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